Chapter 3

Quantum-Chemical Methods for Accurate Theoretical Thermochemistry

Krishnan Raghavachari

Electronic and Photonic Materials Physics Research, Agere Systems, Murray Hill, NJ

07974, U.S.A.

Larry A. Curtiss

Materials Science and Chemistry Divisions, Argonne National Laboratory, Argonne, IL

60439, U.S.A.

1.INTRODUCTION

The first-principles evaluation of the binding energies of molecules to chemical accuracy is one of the most challenging problems in computational quantum chemistry. Dramatic progress has been made in this regard in the last two decades and the rigorous demands placed on the theoretical methods to achieve this goal are now well understood. In principle it is now known how to compute the binding energies and other thermochemical properties of most molecules to very high accuracy [1-10]. This can be achieved by using very high levels of correlation, such as that obtained with coupled cluster [CCSD(T)] [11] or quadratic configuration interaction [QCISD(T)] [12] methods, and very large basis sets containing high angular momentum functions. The results of these calculations are then extrapolated to the complete basis set limit and corrected for some smaller effects such as core-valence and relativistic effects. Unfortunately, this approach is limited to small molecules because of the scaling (with respect to the number of

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©2001 Kluwer Academic Publishers. Printed in the Netherlands.

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basis functions N) of the correlation methods and the need for very large basis sets.

An alternative approach applicable for larger molecules is to use a series of high-level correlation calculations [e.g., QCISD(T), MP4 [13], or CCSD(T)] with moderate sized basis sets to approximate the result of a more expensive calculation. The Gaussian-n series [14-30] exploits this idea to predict thermochemical data. In addition, molecule-independent empirical parameters are used in these methods to estimate the remaining deficiencies in the calculations. Such an approach using higher-level corrections (additive parameters that depend on the number of paired and unpaired electrons in the system) has been quite successful, and the latest version, Gaussian-3 (G3) theory [21], achieves an overall accuracy of 1 kcal/mol for the G2/97 test set [24, 25]. Petersson et al. [31] have developed a related series of methods, referred to as complete basis set (CBS) procedures, for the evaluation of accurate energies of molecular systems. The central idea in the CBS methods is an extrapolation procedure to determine the projected second-order (MP2) energy in the limit of a complete basis set. Several empirical corrections, similar in spirit to the higher-level correction used in the Gaussian-n series, are added to the resulting energies in the CBS methods to remove systematic errors in the calculations. Another approach to calculation of thermochemical data that has been proposed is scaling of the calculated correlation energy using multiplicative parameters [32-36] determined by fitting to experimental data. Finally, hybrid density functionals are being used increasingly to predict the thermochemistry of molecules with reasonable accuracy [24-26].

In this chapter, we review the elements of G3 theory and related techniques of computational thermochemistry. This review is restricted almost exclusively to the techniques that we have developed and the reader is referred to the remaining chapters in this volume for other complementary approaches. An important part of the development of such quantum-chemical methods is their critical assessment on test sets of accurate experimental data. Section 3.2 provides a brief description of the comprehensive G3/99 test set [26] of experimental data that we have collected. Section 3.3 discusses the components of G3 theory as well as the approximate versions such as G3(MP3) [22] and G3(MP2) [23], and their performance for the G3/99 test set. The G3S method [29] that includes multiplicative scale factors is presented in section 3.4 along with other related variants. Section 3.5 discusses the recently developed G3X method [30] that corrects for most of the deficiencies of G3 theory for larger molecules. The performance of these methods is compared to

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that of some of the popularly used density functionals in section 3.6. Finally, conclusions are drawn in section 3.7.

2.THE G3/99 TEST SET

Critical documentation and assessment of quantum-chemical models is essential for such methods to become predictive tools for chemical investigation. We have assembled a large test set of good, credible experimental data to perform such assessments [24-26]. The current test set, referred to as G3/99 [26], contains 376 energies (222 enthalpies of formation, 88 ionization energies, 58 electron affinities, and 8 proton affinities) that are known experimentally [37-39] to an accuracy of better than It includes three subsets of energies, G2-1, G2-2, and G3-3. The G2-1 subset (original G2 test set) includes the enthalpies of formation for only very small molecules containing 1 - 3 heavy atoms (systems such as and ), whereas G2-2 includes medium-sized molecules containing 3 - 6 heavy atoms (systems such as etc.). It also includes ionization energies on some larger molecules such as substituted benzenes. The two subsets, G2-1 and G2-2, are together referred to as G2/97 and contain 301 test energies [24, 25]. The G3-3 test set [26] comprises 75 new enthalpies of formation for molecules that are, on average, larger (containing 3 - 10 heavy atoms). The largest molecule in the G3-3 test set contains ten non-hydrogen atoms (e.g., naphthalene or azulene). It also includes some larger hypervalent molecules such as or that provide a challenge for many theoretical models.

The 222 enthalpies of formation included in the G3/99 test set contain a wide variety of molecules with many different kinds of bonds. They are conveniently classified into subgroups of molecules. They include 47 molecules containing non-hydrogen atoms, 38 hydrocarbons, 91 substituted hydrocarbons, 15 inorganic hydrides, and 31 open-shell radicals. Together, they provide a comprehensive assessment of new theoretical models in a wide variety of bonding environments.

The collection of such a large set of experimental data provides many challenges. All the experimental values that are included have a quoted uncertainty of less than 1 kcal/mol [37-39]. However, the evaluation of the experimental uncertainties is difficult or impossible in many cases. It is possible that some of the included values may turn out to be incorrect. For example, the G2/97 test set originally comprised 302 energies, but the enthalpy of formation of has been deleted because a new experimental upper limit [40] has been reported that casts doubt on

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the value used in the G2/97 test set. In addition, on the basis of theoretical evidence, a few other enthalpies of formation (vinyl chloride, and ) and one ionization energy may not be as accurate as cited experimentally [41,42]. In our analysis, we have chosen not to throw out experimental data unless there is new experimental evidence that warrants it. Another important factor is that the calculation of the enthalpies of formation for molecules requires the experimental atomic enthalpies of formation. Two of these (B and Si) have significant uncertainties and some authors have suggested the use of ”theoretical” atomic enthalpies of formation for Si and B in the calculation of molecular enthalpies of formation [43-45]. We have consistently used experimental values for all elements, despite the uncertainty in the Si and B values. The reason that we do not use these ”theoretical” atomic enthalpies is that they are derived in part from an experimental molecular enthalpy that is part of the test set, which may bias the assessment process [46]. If the accuracy of theory improves and becomes demonstrably better than that of experiment, theoretical values may be included in the future to assemble test sets of molecules for critical assessment.

3.GAUSSIAN-3 THEORY

Gaussian-3 theory, like its predecessor Gaussian-2 (G2) theory [17], is a composite technique in which a sequence of well-defined ab initio molecular orbital calculations [47] is performed to arrive at a total energy of a given molecular species. It was designed to correct some of the deficiencies of G2 theory for systems such as halogen-containing molecules, unsaturated hydrocarbons, etc. It also contains important physical effects, such as core-valence correlation and spin-orbit contributions, that were not included in G2 theory. G3 theory is computationally less demanding than G2 theory though it is significantly more accurate. The detailed steps involved in G3 theory are as follows:

1.An initial equilibrium structure is obtained at the Hartree-Fock (HF) level with the 6-31G(d) basis [47]. Spin-restricted (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. The HF/6-31G(d) equilibrium structure is used to calculate harmonic frequencies, which are then scaled by a factor of 0.8929 to take account of known deficiencies at this level [48]. These frequencies are used to evaluate the zero-point energy and thermal effects.

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2.The equilibrium geometry is refined at the MP2(fu)/6-31G(d) level, using all electrons for the calculation of correlation energies. This is the final equilibrium geometry in the theory and is used for all single-point calculations at higher levels of theory in step 3. Except where otherwise noted by the symbol (fu), these subsequent calculations include only valence electrons in the treatment of electron correlation.

3.A series of single-point energy calculations is carried out at higher levels of theory. The first higher-level calculation is the complete fourth-order Møller-Plesset perturbation theory [13] with the 6- 31G(d) basis set, i.e. MP4/6-31G(d). For convenience of notation, we represent this as MP4/d. This energy is then modified by a series of corrections from additional calculations:

(a)A correction for correlation effects beyond fourth-order perturbation theory using the quadratic configuration interaction (QCI) method [12],

(b)A correction for diffuse functions,

where plus denotes the 6-31+G(d) basis set [47].

(c) A correction for higher polarization functions on nonhydrogen atoms and p-functions on hydrogens,

where 2df,p denotes the polarized 6-31G(2df,p) basis set [47].

(d) A correction for larger basis set effects and for the non-additivity caused by the assumption of separate basis set extensions for diffuse functions and higher polarization functions,

The largest basis set, denoted as G3Large [21] includes some core polarization functions as well as multiple sets of valence polarization functions. It should be noted that MP2 calculation with the largest basis set in Eq. (3.4) is carried out at the MP2(fu) level.

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This is done to take account of core-related correlation contributions to total energies.

4.Spin-orbit correction is included for atomic species only. The spin-orbit correction is taken from experiment [49] where available and accurate theoretical calculations [50] in other cases. These corrections are particularly important for halide-containing systems [24]. Molecular spin-orbit corrections are not included in G3 theory.

5.A ”higher-level correction” is added to take into account remaining deficiencies in the energy calculations: is given by

for molecules and for atoms (including atomic ions), where the and are the numbers of and valence electrons, respectively, with The number of valence electron pairs corresponds to Thus, A is the correction for pairs of valence electrons in molecules, B is the correction for unpaired electrons in molecules, C is the correction for pairs of valence electrons in atoms, and D is the correction for unpaired electrons in atoms. The use of different corrections for atoms and molecules can be justified, in part, by noting that effects of basis functions with higher angular momentum are likely to be of more importance in molecules than in atoms. The A, B, C, D values are chosen to give the smallest average absolute deviation from experiment for the G2/97 test set. For G3 theory,

6.Finally, the total energy at 0 K (”G3 energy”) is obtained by adding all the individual energy corrections in an additive manner,

The G3 energy can also be represented more fully as

The final total energy is effectively at the QCISD(T,fu)/G3Large level if the additivity approximations used work well. The validity of

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such approximations has been previously investigated for G2 theory on the G2-1 subset of G2/97 and found to be satisfactory [19].

The correlation methods in G3 theory are still computationally demanding and it is of interest to find modifications to reduce the computational requirements. Two approximate versions of G3 theory have been proposed to make the methods more widely applicable. The first is G3(MP3) [22] that eliminates the expensive MP4/2df,p calculation by evaluating the larger basis set effects at the MP3 level. It also eliminates the MP4/plus calculation,

The second is G3(MP2) theory [23] that evaluates the larger basis set effects at the MP2 level, similar to the successful G2(MP2) theory,

In G3(MP2) theory, the MP2(fu)/G2Large calculation of G3 is replaced with a frozen core calculation with the G3MP2Large basis set [23] that does not contain the core polarization functions of the G3Large basis set.

The enthalpies of formation for most molecules in the G2/97 and G3/99 test sets have been measured at 298 K. In order to compare with experiment, the heats of formation for molecules are calculated using a procedure described in detail previously [24]. Briefly, thermal corrections (298 K) are first evaluated using the calculated vibrational frequencies and standard statistical-mechanical methods [51]. The calculated total energies of the given molecule and its constituent atoms are used to evaluate its atomization energy. This value is then used along with the thermal corrections and the known experimental enthalpies of formation for the atomic species [21, 38] to calculate the enthalpy of formation for the molecule (298 K). The electron affinities are calculated as the difference in total energies at 0 K of the anion and the corresponding neutral at their respective MP2(fu)/6-31G(d) optimized geometries. Likewise, the ionization potentials are calculated as the difference in total energies at 0 K of the cation and the corresponding neutral at their respective MP2(fu)/6-31G(d) optimized geometries. The Gaussian 98 computer

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program is used for the calculations [52]. Many of the G3 techniques have been implemented in this computer program.

The performance of G3, G3(MP3), and G3(MP2) theories for the energies in the G2/97 and G3/99 test sets is summarized in Table 3.1. Overall, the mean absolute deviations increase slightly for the G3/99 test set compared to that of the G2/97 test set. The mean absolute deviation of G3 theory increases from 1.01 kcal/mol to 1.07 kcal, that of G3(MP3) theory increases from 1.21 kcal/mol to 1.27 kcal/mol, and that of G3(MP2) theory remains at 1.31 kcal/mol. This increase in the mean absolute deviation is primarily due to large deviations in the calculated enthalpies of formation of some of the non-hydrogen species in the expanded test set. In particular, the mean absolute deviation of

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3.24 kcal/mol for the 13 non-hydrogens in the G3-3 subset is nearly twice that of the 34 non-hydrogens in the G2/97 set (1.68 kcal/mol). Especially large deviations in the G3 data occur for (-6.22 kcal/mol), (-7.05 kcal/mol), (-5.14 kcal/mol), (-4.15 kcal/mol), and (-4.37 kcal/mol). Among these, is an unusually strained molecule with a bond angle of 60°. The remaining systems are hypervalent. Overall, the G3 deviations for nearly all of the new non-hydrogen

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species are negative, indicating underbinding. Part of the error for these species is due to the use of MP2/6-31G(d) geometries. For example, when experimental geometries are used, the deviation for decreases from -6.22 to -3.42 kcal/mol, that for decreases from -7.05 to -4.93 kcal/mol, and that for falls from -5.14 to -2.53 kcal/mol. The remainder of the discrepancies for non-hydrogen systems is mostly due to basis set deficiencies. The mean absolute deviations for the other types of molecules in the G3/99 test set are similar to those in the G2/97 test set.

As mentioned earlier, G3 theory was designed to correct for some of the deficiencies in G2 theory. The histograms in Fig. 3.1 show the range of deviations of G2 and G3 theories from experiment for the G2/97 test set. Nearly 88 % of the G3 deviations fall within the range of -2.0 to 2.0 kcal/mol. This is substantially better than G2 theory for which about 74 % of the deviations fall in this range. In addition to improving the accuracy, the use of the 6-31G(d)-based calculations in G3 theory substantially decreases the computer time as well as disk space requirements relative to G2 theory [which uses the larger 6-311G(d,p)- based calculations]. For example, the G3 calculation on benzene is nearly twice as fast as the analogous G2 calculation.

As proposed originally, G3 theory is applicable only to molecules containing atoms of the first (Li - F) and second (Na - Cl) rows of the periodic chart. It has recently been extended [53] to molecules containing the third-row non-transition elements K, Ca, and Ga - Kr. Basis sets compatible to those used in G3 theory for molecules containing firstand second-row atoms have been derived. The G3 mean absolute deviation from experiment for a set of 47 test energies containing these elements is 0.94 kcal/mol. This is a substantial improvement over G2 theory for the third row, which has a mean absolute deviation of 1.43 kcal/mol for the same set [54, 55]. Variations of G3 theory based on reduced orders of perturbation theory that are similar to those for G2 theory [56] have also been reported [53]. G3(MP2) theory for third-row molecules has a mean absolute deviation from experiment of 1.30 kcal/mol, and is significantly more accurate than G2(MP2). The G3 method based on third-order perturbation theory, G3(MP3), has an average absolute deviation of 1.24 kcal/mol. In addition, these methods have been assessed on a set of molecules containing K and Ca for which the experimental data is not accurate enough for them to be included in the test set [53]. Results for this set indicate that G3 theory performs significantly better than G2 for molecules containing Ca.

Other variants of G3 theory have been proposed that use alternate geometries, zero-point energies, or higher-order correlation methods. G3